Have you tried the Audacity noise removal tool under "effects". I get a peculiar hum when I connect my amp (Peavey Vyper 15) directly. You just sample a couple of seconds of silence (with the hum) at the beginning and the software removes it throughout the recording. As well as it works though I usually just stick a mike in front of my amp and record that way because I can't figure out how to record it and hear it at the same time the other way. Still learning the software myself, so it's probably something I'm doing wrong.
I love Audacity and playing to the backing tracks. Being able to set loops to repeat measures and change the tempo without changing the pitch is such a help...sooner or later I'll record a version that I'm not too embarrassed to share.